Research Article Integral Equation Analysis of EM ...downloads.hindawi.com/journals/ijap/2015/274307.pdf · Research Article Integral Equation Analysis of EM Scattering from Multilayered

Research ArticleIntegral Equation Analysis of EM Scattering fromMultilayered Metallic Photonic Crystal Accelerated withAdaptive Cross Approximation

Jianxun Su1 Zengrui Li1 Xujin Yuan2 Yaoqing (Lamar) Yang3 and Junhong Wang4

1Electromagnetic Laboratory Communication University of China Beijing 100024 China2Science and Technology on Electromagnetic Scattering Laboratory Beijing 100854 China3Department of Computer and Electronics Engineering University of Nebraska-Lincoln Lincoln NE 68182 USA4Institute of Lightwave Technology Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Jianxun Su sujianxun jlgx163com

Received 6 March 2015 Accepted 8 July 2015

Academic Editor Luis Landesa

Copyright copy 2015 Jianxun Su et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A space-domain integral equation method accelerated with adaptive cross approximation (ACA) is presented for the fast andaccurate analysis of electromagnetic (EM) scattering from multilayered metallic photonic crystal (MPC) The method directlysolves for the electric field in order to easily enable the periodic boundary condition (PBC) in the spatial domain The ACA isa purely algebraic method allowing the compression of fully populated matrices hence its formulation and implementation areindependent of integral equation kernel (Greenrsquos function)Therefore the ACA is very well suited for accelerating integral equationanalysis of periodic structurewith the integral kernel of the periodicGreenrsquos function (PGF)The computation of the spatial-domainperiodic Greenrsquos function (PGF) is accelerated by the modified Ewald transformation such that the multilayered periodic structurecan be analyzed efficiently and accurately An effective interpolation method is also proposed to fast compute the periodic Greenrsquosfunction which can greatly reduce the time of matrix filling Numerical examples show that the proposed method can greatly savethe frequency sweep time for multilayered periodic structure

1 Introduction

Photonic crystals [1] are periodic structures of great interestfor their applications both in themicrowave region and in theoptical range The main feature is the presence of frequencybands wherein the waves are highly attenuated and cannotpropagate It results from the removal of degeneracies ofthe free-photon states at the Bragg planes provoked by theperiodicity which produces forbidden frequency gaps so-called photonic band gaps (PBGs) This property is exploitedin the electromagnetic and optical applications

Many numerical methods for computing such bandstructures have been adapted from solid-state physics Theyinclude but are not restricted to plane-wave expansions[2] the Korringa-Kohn-Rostoker method [3] and shellmethodologies [4] Over the past two decades finite dif-ference time domain (FDTD) [5] has been widely used in

computational optics and photonics Recent developmentsinclude specialized versions of the finite element method(FEM) [6] and the flexible local approximation method[7] FDTD and FEM can cope with arbitrary complicatedshapes andmaterials However the radiation into unboundedregions requires either absorbing boundary conditions orperfectly matched layers and special measures need to betaken into account when employing these conditions tothe scattered field formulation For open-region problemsintegral equationmethod of moments (IEMoM) has a greatadvantage

The ACA is a purely algebraic method which is inde-pendent of the kernel The ACA was originally introduced in2000 by Bebendorf [8] and since then has been successfullyapplied for integral formulations involved electromagneticscattering [9 10] The integral kernel is a free space Greenrsquosfunction In addition it is noted that most photonic crystal

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 274307 7 pageshttpdxdoiorg1011552015274307

2 International Journal of Antennas and Propagation

applications deal with two-dimensional (2D) structures thatare invariant along a longitudinal axis and periodic in thetransverse plane [11] The manufacture of a 2D photoniccrystal structure is easier than that of three-dimensional (3D)one [12]

In this paper a space-domain integral equation acceler-ated with ACA is presented for the fast and accurate analysisof electromagnetic scattering from 2D and 3D multilayeredperiodic structures The ACA is first introduced for surfaceintegral equation with the kernel of periodic Greenrsquos function(PGF) The ACA can be applied to compress impedancematrix blocks and accelerate matrix-vector multiplication(MVP) for reducing computational complexity A modifiedEwald transformation is proposed to efficiently computethe PGFs Compared with traditional Ewald method it hasless computational cost and can eliminate the imbalance oftwo Ewald sequences for multilayered periodic structureAn efficient interpolation method is also proposed to fastcompute the PGFs and then accelerate impedance matrixfilling Our proposed method can effectively reduce thefrequency sweep time for multilayered periodic structureinvolving a lot of unknowns

2 Formulation

Figure 1 shows the multilayered double periodic structurewith identical metallic objects of arbitrary shape periodicallyrepeated in the 119909119910-plane 997888119886 1 and

997888119886 2 are the primitive lattice

vectors 119860 = |997888119886 1 times

997888119886 2| is the cross-sectional area of the unit

cell 997888119864119894

= (120579 cos120572 + 120593 sin120572)1198640 exp(minus119895120573119896 sdot

997888119903 ) is the incident

electric field 120572 is the polarization angle The wave number is120573 = 120596radic120583120576 and 119896 = minus(119909 sin 120579119894 cos120593119894+119910 sin 120579119894 sin120593119894+ cos 120579119894)The wave vector is

997888

119896 = 120573119896 and phase shift

997888

11989611990500 =

997888

119896 |119905 The

array is assumed to be periodic in the 119909119910-plane and the cell ofthe array is obtained by shifting the cell through the relation997888120588119898119899= 119898sdot

997888119886 1+119899sdot

997888119886 2

997888120588119898119899

is defined as the translation vectorof the lattice

21 Integral Equation In this paper we consider the metallicphotonic crystal Let 119878 be the surface of a metallic object Byenforcing the boundary conditions that the total tangentialelectric field is zero on the perfect electric conductor (PEC)surface 119878 the electric field integral equation (EFIE) is givenas [13]

119895120596120583 [int[

997888

119869119904(997888119903

1015840

) +

11205732nabla1015840sdot

997888

119869119904(997888119903

1015840

)nabla]

sdot 119866119901(997888119903 997888119903

1015840

) 1198891199041015840]

119905

=

997888

119864

inc119905

(1)

where 119866119901is a doubly periodic Greenrsquos function The Rao-

Wilton-Glisson (RWG) basis functions on triangular ele-ments are employed to discretize the electric current [14] Inaddition the special basis and test functions derived fromRWG basis function are introduced to ensure current con-tinuity across the periodic boundaries [15] The applicationof the Floquet-Bloch theorem reduces the computational

x

y

zPlane wave120579

120593

a1

a2

Figure 1 Multilayered periodic structure with a general skewedlattice

domain of infinite periodic structures to a single unit cell butleads to the numerical evaluation of very slowly convergingseries

22 Modified Ewald Transformation We propose the follow-ing strategy to compute the spatial periodic Greenrsquos functionin an arbitrary point 997888119906 = 997888119903 minus 997888119903

1015840

If 119906119911= |119911 minus 119911

1015840| gt 05radic119860 PGF is computed by its spectral

representation with exponential convergence and thus noacceleration technique is required [16] Consider

119866119901(997888119903 997888119903

1015840

) =

infin

sum

119898=minusinfin

infin

sum

119899=minusinfin

119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840

)

2119895119860119896119911119898119899

119890minus119895119896119911119898119899|119911minus1199111015840| (2)

where

997888119903 =

997888120588 +119911 sdot

997888

119896119905119898119899

=

997888

11989611990500 +

2120587119860

[119898 (997888119886 2 times ) + 119899 ( times

997888119886 1)]

(3)

is the reciprocal lattice vector Consider

119896119911119898119899

=radic11989620 minus

997888

119896119905119898119899

sdot

997888

119896119905119898119899

(4)

where Re(119896119911119898119899

) ge 0 Im(119896119911119898119899

) le 0If 119906119911= |119911 minus 119911

1015840| lt 05radic119860 PGF is computed by the

representation of Ewald transformation [17] which are twoexponential converging series Consider

119866119901(997888119903 997888119903

1015840

) = 1198661199011 (

997888119903 997888119903

1015840

) +1198661199012 (

997888119903 997888119903

1015840

) (5)

International Journal of Antennas and Propagation 3

Primary source

Field

Image sources

Image sources

(a)

(minus1 0) (0 0) (1 0)

(minus1 minus1) (0 minus1) (1 minus1)

(minus1 1) (0 1) (1 1)

(b)

Figure 2 (a) Periodic image sources in triangular boundary mesh (b) Corresponding lattice numbers (0 0) lattice is the primary unit cell

where

1198661199011 (

997888119903 997888119903

1015840

) =

infin

sum

119898=minusinfin

infin

sum

119899=minusinfin

119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840

)

4119895119860119896119911119898119899

sdot [119890minus119895119896119911119898119899(119911minus1199111015840) erfc(

119895119896119911119898119899

2119864minus (119911 minus 119911

1015840) 119864)

+ 119890119895119896119911119898119899(119911minus1199111015840) erfc(

119895119896119911119898119899

2119864+ (119911 minus 119911

1015840) 119864)]

(6)

1198661199012 (

997888119903 997888119903

1015840

) =

infin

sum

119898=minusinfin

infin

sum

119899=minusinfin

119890minus11989599788811989611990500 sdot997888120588119898119899

4120587119877119898119899

sdot real [1198901198951198960119877119898119899 erfc(119877119898119899119864+

119895119896

2119864)]

(7)

where erfc(119911) is the complementary error function of com-plex argument and 119877

119898119899= |997888119903 minus

997888119903

1015840

minus997888120588119898119899|

By analyzing the asymptotic behavior of the series termsthe approximation to the optimal value for general skewedlattices is given by

119864 = max(radic 120587

119860

119896

2119867) (8)

where1198672 is the maximum exponent permitted In almost allpractical cases it is sufficient to include only nine terms in (6)and (7) (ie the summation limits are fromminus1 to +1) inwhichthe error is usually less than 01

For multilayered periodic structure when 119906119911= |119911 minus 119911

1015840| gt

05radic119860 the PGFs are computed by their spectral represen-tation rather than direct Ewald transformation In doing sothere are two advantages

(1) The spectral representation just requires evaluation ofthe exponential function which is much simpler than

the computation of complex-argument complemen-tary error function (erfc) Compared with erfc(119911)the computational complexity of the exponentialfunction exp(119911) can be ignored

(2) In addition it can eliminate the imbalance of twoEwald sequences when 119906

119911is too large For the multi-

layered structure the complementary error functionsin the first terms (119898 = 119899 = 0) of both Ewald serieswill take large imaginary arguments when 119906

119911is too

largeThese two terms will have very large values thatare approximately equal in amplitude but of oppositesigns so that summing them up will lead to a severeloss of accuracy due to the finite machine precision

23 Singularity Extraction When the distance between theobservation point 997888119903 and the source point 997888119903

1015840

or one of itsperiodic extensions is electrically small the contribution toGreenrsquos function is essentially quasistatic Usually the centerdistance between source and field triangles is less than 01120582then both are considered to be ldquonear interactionrdquo Singularterms in (5) and (7) often arise for example from one ofnine spatial terms 119877

119898119899with119898 119899 = minus1 0 or +1 depending on

which term represents the source nearest to the observer 997888119903 By extracting its singularity [18] the PGF can be expressedas 119866119901= 1198661199011+ [1198661199012minus 1(4120587119877)] + 1(4120587119877) The singularity

subtraction technique [19] used in this paper can compute thesingularity of |997888119903 minus 997888119903

1015840

|119899(119899 ge minus3) type

Supposing the source point nearest to the observer pointappears at (1 0) term rather than (0 0) term as shown inFigure 2 the primary source point can be transferred to (1 0)lattice Then the singular term always appears at (0 0) term11986600

1199012 The singularity of 11986600

1199012can be extracted as

119866001199012 (

997888119903 997888119903

1015840

) = 119865001199012 (

997888119903 997888119903

1015840

) +

14120587119877

(9)


o

a2

a1

Gn119909n119910n119911Gn119909+1n119910n119911

Gn119909+1n119910+1n119911Gn119909n119910+1n119911

Gn119909+1n119910n119911+1

Gn119909n119910+1n119911+1

Gn119909n119910n119911+1

Gn119909+1n119910+1n119911+1

dx

dydz

G(u)

Figure 3 Depiction of trilinear interpolation of PGF

where 119865001199012

is the nonsingular part

119865001199012 (

997888119903 997888119903

1015840

) =

14120587119877

sdot real119890minus1198951198960119877 sdot erfc(119877119864minus11989511989602119864

)minus erfc(minus11989511989602119864

)

(10)

As 119877 rarr 0 the result of applying Lrsquo Hopitalrsquos rule on (10) is asfollows

lim119877rarr 0

119865001199012 (

997888119903 997888119903

1015840

)

=

12120587

sdot real [minus1198951198960 erfc(minus11989511989602119864

)minus

2119864radic120587

119890119896204119864

2]

(11)

24 Interpolation of PGF PGFs are very smooth andamenable to interpolation Fast interpolation of PGFs cangreatly reduce the time of matrix filling The reason is thatinterpolating the PGF values from the table is much fasterthan directly computing the Ewald sums (since this has tobe done for every pair of source and field triangles in thenumerical integration using cubature formulas) Trilinearinterpolation [20] suited for the calculation of complex-valued function with very high efficiency is applied hereAs shown in Figure 3 the cuboid grouping the unit cell isequally divided intomany subgrids with side length 119871 Yellowpoints are the precomputed PGFs at the corner of the subgridThen the three-dimensional PGF values can be obtained bytrilinear interpolation

119866(997888119906) = (1minus120572

119909) (1minus120572

119910) (1minus120572

119911) 119866119899119909+1119899119910+1119899119911+1

+ (1minus120572119909) (1minus120572

119910) 120572119911119866119899119909+1119899119910+1119899119911+2

+ (1minus120572119909) 120572119910(1minus120572

119911) 119866119899119909+1119899119910+2119899119911+1

+ (1minus120572119909) 120572119910120572119911119866119899119909+1119899119910+2119899119911+2

+120572119909(1minus120572

119910) (1minus120572

119911) 119866119899119909+2119899119910+1119899119911+1

+120572119909(1minus120572

119910) 120572119911119866119899119909+2119899119910+1119899119911+2

+120572119909120572119910(1minus120572

119911) 119866119899119909+2119899119910+2119899119911+1

+120572119909120572119910120572119911119866119899119909+2119899119910+2119899119911+2

(12)

where 120572119909= 119889119909119871 120572119910= 119889119910119871 120572119911= 119889119911119871

The relative error 120576 obtained by interpolating the PGFs isdefined as

120576 =

100381610038161003816100381610038161003816100381610038161003816

119866intminus 119866

ex

119866ex

100381610038161003816100381610038161003816100381610038161003816

(13)

where 119866int and 119866

ex are the interpolated and the exactdata respectively The exact data are obtained by the aboveaccelerating method with an accuracy 120576 = 10minus7 The relativeerrors are less than 120576 = 10minus7 and have therefore beenneglected

Take the multilayered structure with orthogonal latticesas an example The relative error is shown in Figure 4 forthe interpolation of PGF involving a series of interpolationpoints through the trilinear interpolation method withvarious values of 119873int In Figure 4 positions (A D) arethe sampling points for interpolation A very high level ofaccuracy is easily obtained with a relatively small number ofinterpolation points The side length of the subgrid is usuallyfixed to 002120582 and the higher interpolation accuracy (120576 lt

10minus4) can be obtained

3 ACA Technique

Let the119898times119899 rectangular matrix119885119898times119899

represent the couplingbetween twowell-separated groups in theMoMcomputationThe ACA algorithm aims to accurately approximate 119885

119898times119899by

a low-rank matrix 119885119898times119899

In particular the ACA algorithm


o

B

C

D

AB

CD

A

10 20 30 40 50 60 70 80 90 100 1101E minus 7

1E minus 6

1E minus 5

1E minus 4

1E minus 3

Rela

tive e

rror

120576Nint

a1

a2

Figure 4 The relative error of a series of points Reference structure primitive vector |9978881198861| = |

9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896

119909=

119896 cos(45∘) 119896119910= 119896 sin(30∘) Accuracy reached interpolatingPGFs through trilinear interpolationwith a 3Dgrid of119873inttimes119873inttimes2119873int equispaced

points in minus055120582 lt Δ119909 lt 055120582 minus055120582 lt Δ119910 lt 055120582 0 lt Δ119911 lt 22120582 The data plotted refers to the straight line from (015120582 015120582 0) to(015120582 015120582 22120582) The number of equally spaced sampling points is 4

constructs the approximate matrix 119885119898times119899

through a productform Namely

119885119898times119899

= 119880119898times119903

119881119903times119899

=

119903

sum

119894=1119906119894

119898times1V119894

119899times1 (14)

where 119903 is the rank of 119885119898times119899

or the effective rank of the matrix119885119898times119899

119880119898times119903

and 119881119903times119899

are two dense rectangular matrices119906119894

119898times1 is the column 119894th of 119880 and V119894119899times1 is the row 119894th of 119881

respectively The goal of ACA is to achieve10038171003817100381710038171003817119885119898times119899

minus119885119898times119899

10038171003817100381710038171003817119865le 120576

1003817100381710038171003817119885119898times119899

1003817100381710038171003817119865

(15)

for a given tolerance 120576 Note that in this paper sdot refers tothe matrix Frobenus norm [9]

The ACA technique is numerically based on the principleof the approximation above but does not need to knowall terms of 119885

119898times119899 Indeed only 119903 rows and 119903 columns

are calculated during the algorithm allowing a significantreduction in the number of integral computations to performTherefore not only is the memory requirement reduced butthe assembly time is also saved Yet if we agree to deal withtheMVP119880

119898times119903119881119903times119899119887119899times1 instead of the exact value of119885119898times119899119887119899times1

the number of operations needed is also remarkably reducedby performing 119880

119898times119903(119881119903times119899119887119899times1) Thus if an iterative solver is

used one can accelerate matrix vector multiply and save onceagain the computation time [10]

The compression rate 120578 of matrix block 119885119898times119899

whichdenotes the memory saved by using ACA can be expressedas

120578 () = (1minus 119903 sdot (119898 + 119899)

119898 sdot 119899

) sdot 100 (16)

Obviously the approximation byACA approach above is use-ful only if 119903 lt (12) sdotmin119898 119899 where 119885

119898times119899presents similar

values For the electromagnetic application the effective rank119903 lt min(119898 119899) Therefore instead of storing entire 119898 times 119899

entries the algorithm only requires storing (119898+119899)times119903 entries

4 Numerical Results

The EFIE and ACA approaches to PEC are implementedusing Fortran-95 language Numerical experiments run ona PC with Intel i5-2400 31 GHz CPU and 16GB RAMThe resulting impedance matrices are iteratively solved byusing the transpose-free quasi-minimal residual (TFQMR)solver with diagonal preconditioning [21] the relative errortolerance is set to be 10minus3

41 Infinite Metallic Rod As a 2D example we study thescattering properties of a photonic band gap (PBG) structureconsisting of infinite metallic rod [17] Each unit cell consistsof two infinitely long metallic rods The inset of Figure 5 istwo meshed cylinders in a unit cell The axis of each cylinderis along 119909-direction 997888119886

1= 119909 6mm and 997888119886

2= 119910 6mm Each

cylinder has the radius 06mm and length 6mm The spacebetween two cylinders is 6mm Each cylinder was discretizedinto 494 triangle facets with 741 unknowns to ensure accurateresults obtained in the entire frequency band Figure 5 showsthe magnitude of the transmission and reflection coefficientsfor the Floquet TM

00mode with the electric field oriented

in the 119909-direction Our results agree well with the referenceresults obtained using 2D IE approach [15] The resultsconverged for 9 terms in both Ewald sums and there are 1482unknowns for the geometry The time for matrix filling andsolving is 058 s and 021 s per frequency point respectivelyInACA the box size is 6times6times6mm3 that is each box includes


minus50

minus40

minus30

minus20

minus10

0

Refl

and

Tran

s co

effici

ents

(dB)

Frequency (GHz)

Refl Trans Refl [15]

o

1

2

Unit cell

10 50403020

a1

a2

Figure 5 Reflection and transmission coefficient of two-layer rodPBG

20 4540353025minus80

minus60

minus40

minus20

0

Tran

simiss

ion

(dB)

Frequency (GHz)

5-layer10-layer21-layer

Layer

1

2

3

N minus 1

N

Figure 6 Transmission coefficient of multilayered PBG

one cylinder If the trilinear interpolation method is notused the time ofmatrix filling would increase dramatically to1286 s Therefore trilinear interpolation of PGFs can reducethe time of matrix filling by more than 995

Next the number of layers increases from 2 to 5 10 and21 respectively Figure 6 shows the transmission coefficientofmultilayered structureThe transmission coefficient curvesexhibit the typical resonances of photonic band gapmaterialsIt can be seen that the null of transmission becomes deeperas the layer number increasesThe total solution times for perfrequency point are 51 s 124 s and 325 s respectively

The 21-layer structure is meshed into 10374 triangle facetswith 15561 unknownsThe compression rate 120578 ofmatrix blockis shown in Table 1 The times for matrix filling and matrixsolving are 238 s and 87 s per frequency point respectively

10 201816141200

02

04

06

08

10

Refle

ctio

n

4-layer 8-layer

16-layer 16-layer [Ansys HFSS]

w

d

h

120582 (120583m)

Figure 7 Reflection spectra for multilayered woodpile MPC

Table 1 Compression rate 120578 () of matrix block

11988511

11988512

11988513

11988514

11988515

11988516

11988517

mdash 2078 864 540 594 567 48611988518

11988519

119885110

119885111

119885112

119885113

119885114

621 567 405 351 378 459 432119885115

119885116

119885117

119885118

119885119

119885120

119885121

486 540 432 378 405 324 351mdash denotes the matrix block calculated directly by MoM

The corresponding times of direct solution by MoM withoutACA acceleration are 1876 s and 752 s per frequency pointrespectively Obviously ACA can greatly reduce the totalsolution time especially for frequency sweeping

42 Woodpile A 3D example showing the inset in Figure 7is a woodpile MPC with 119908 = 200 nm ℎ = 300 nm andin-layer lattice constant 119889 = 1 120583m The incident wave is aTE polarized (ie 119864 parallel to the top metallic layer) planewave through the stacking direction The number of layersare 4 8 and 16 respectively Two intersecting metallic pipesare considered to be fully contacting with each other innumerical modelingTheACA box size is 1times1times03 120583m3Thereflection coefficients are shown in Figure 7 Almost the sameresult is obtained for TM polarization Our numerical resultsagree well with that of Ansys HFSS Previous results reportedby other studies [12 22] for a similar kind of the structuralgeometry also validate our calculated results For the 16-layerMPC with 13149 unknowns (8766 triangle facets) the timesformatrix filling and linear system solving are 403 s and 152 sper frequency point respectivelyThe corresponding times ofdirect solution by MoMwithout ACA acceleration are 1323 sand 925 s respectively

5 Conclusion

An integral equation accelerated with adaptive cross approx-imation (ACA) is proposed for the analysis of multilayeredmetallic photonic crystal The ACA is a purely algebraic and


kernel independent algorithm It is verified to be suited foraccelerating integral equation analysis of periodic structureThe ACA can be applied to compress impedance matrixblocks and accelerate MVP for reducing computational com-plexity A modified Ewald transformation and an efficientinterpolation method are also proposed to fast compute theperiodic Greenrsquos function hence that can greatly reducethe time of matrix filling Numerical results confirm thevalidity and accuracy of the proposed method In our nextstage of research ACA will be applied to accelerate hybridfield integral equation for arbitrarily complex multilayeredperiodic structure composed ofmetal and dielectricmaterialswith more unknowns

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work was supported in part by the National ScienceFoundation of China (NSFC) under Grants no 61331002no 61102011 and no 61201082 and in part by ExcellentInnovation Team of CUC under Grant no yxtd201303

References

[1] J D Joannopoulos R D Meade and J N Winn PhotonicCrystals Molding the Flow of Light Princeton University PressPrinceton NJ USA 1995

[2] R D Meade A M Rappe K D Brommer J D Joannopoulosand O L Alerhand ldquoAccurate theoretical analysis of photonicband-gapmaterialsrdquo Physical Review B vol 48 no 11 pp 8434ndash8437 1993

[3] X Wang X-G Zhang Q Yu and B N Harmon ldquoMultiple-scattering theory for electro-magnetic wavesrdquo Physical ReviewB vol 47 no 8 pp 4161ndash4167 1993

[4] J B Pendry ldquoPhotonic structuresrdquo Journal of Modern Opticsvol 41 pp 209ndash229 1994

[5] J A Roden S D Gedney M P Kesler J G Maloney andP H Harms ldquoTime-domain analysis of periodic structuresat oblique incidence orthogonal and nonorthogonal FDTDimplementationsrdquo IEEE Transactions on MicrowaveTheory andTechniques vol 46 no 4 pp 420ndash427 1998

[6] A A Tavallaee and J P Webb ldquoFinite-element modeling ofevanescent modes in the stopband of periodic structuresrdquo IEEETransactions on Magnetics vol 44 no 6 pp 1358ndash1361 2008

[7] I Tsukerman and F Cajko ldquoPhotonic band structure compu-tation using FLAMErdquo IEEE Transactions on Magnetics vol 44no 6 pp 1382ndash1385 2008

[8] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000

[9] K ZhaoM Vouvakis and J-F Lee ldquoThe adaptive cross approx-imation algorithm for accelerated method of moments compu-tations of EMC problemsrdquo IEEE Transactions on Electromag-netic Compatibility vol 47 no 4 pp 763ndash773 2005

[10] A Heldring J M Tamayo C Simon E Ubeda and J M RiusldquoSparsified adaptive cross approximation algorithm for acceler-ated method of moments computationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 1 pp 240ndash246 2013

[11] M Sarnowski T Vaupel V Hansen E Kreysa and H PGemuend ldquoCharacterization of diffraction anomalies in 2-Dphotonic bandgap structuresrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 49 no 10 pp 1868ndash1872 2001

[12] S Y Lin G Arjavalingam andWM Robertson ldquoInvestigationof absolute photonic bandgaps in 2-dimensional dielectricstructuresrdquo Journal of Modern Optics vol 41 pp 385ndash393 1994

[13] W C Chew J M Jin E Michielssen and J Song Fast andEfficient Algorithm in Computational Electromagnetics chapter3 Artech House Boston Mass USA 2001

[14] S M Rao D R Wilton and A W Glisson ldquoElectromagneticscattering by surfaces of arbitrary shaperdquo IEEE Transactions onAntennas and Propagation vol 30 no 3 pp 409ndash418 1982

[15] F-G Hu and J Song ldquoIntegral-equation analysis of scatteringfrom doubly periodic array of 3-D conducting objectsrdquo IEEETransactions on Antennas and Propagation vol 59 no 12 pp4569ndash4578 2011

[16] M G Silveirinha and C A Fernandes ldquoA new accelerationtechnique with exponential convergence rate to evaluate peri-odic Green functionsrdquo IEEE Transactions on Antennas andPropagation vol 53 no 1 pp 347ndash355 2005

[17] K E Jordan G R Richter and P Sheng ldquoAn efficient numericalevaluation of the Greenrsquos function for the Helmholtz operatoron periodic structuresrdquo Journal of Computational Physics vol63 no 1 pp 222ndash235 1986

[18] R D Graglia ldquoOn the numerical integration of the linear shapefunctions times the 3-D Greenrsquos function or its gradient on aplane trianglerdquo IEEETransactions onAntennas andPropagationvol 41 no 10 pp 1448ndash1455 1993

[19] S Jarvenpaa M Taskinen and P Yla-Oijala ldquoSingularitysubtraction technique for high-order polynomial vector basisfunctions on planar trianglesrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 42ndash49 2006

[20] J F Hughes and A V Dam Computer Graphics Principles andPractice Addison-Wesley 3rd edition 2013

[21] Y Saad Iterative Methods for Sparse Linear Systems SIAMSociety for Industrial amp Applied Mathematics 2nd edition2003

[22] J G Fleming S-Y Lin I El-Kady R Biswas andKMHo ldquoAll-metallic three-dimensional photonic crytal with a large infraredbandgaprdquo Nature vol 417 no 6884 pp 52ndash55 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



applications deal with two-dimensional (2D) structures thatare invariant along a longitudinal axis and periodic in thetransverse plane [11] The manufacture of a 2D photoniccrystal structure is easier than that of three-dimensional (3D)one [12]

In this paper a space-domain integral equation acceler-ated with ACA is presented for the fast and accurate analysisof electromagnetic scattering from 2D and 3D multilayeredperiodic structures The ACA is first introduced for surfaceintegral equation with the kernel of periodic Greenrsquos function(PGF) The ACA can be applied to compress impedancematrix blocks and accelerate matrix-vector multiplication(MVP) for reducing computational complexity A modifiedEwald transformation is proposed to efficiently computethe PGFs Compared with traditional Ewald method it hasless computational cost and can eliminate the imbalance oftwo Ewald sequences for multilayered periodic structureAn efficient interpolation method is also proposed to fastcompute the PGFs and then accelerate impedance matrixfilling Our proposed method can effectively reduce thefrequency sweep time for multilayered periodic structureinvolving a lot of unknowns

2 Formulation

Figure 1 shows the multilayered double periodic structurewith identical metallic objects of arbitrary shape periodicallyrepeated in the 119909119910-plane 997888119886 1 and

997888119886 2 are the primitive lattice

vectors 119860 = |997888119886 1 times

997888119886 2| is the cross-sectional area of the unit

cell 997888119864119894

= (120579 cos120572 + 120593 sin120572)1198640 exp(minus119895120573119896 sdot

997888119903 ) is the incident

electric field 120572 is the polarization angle The wave number is120573 = 120596radic120583120576 and 119896 = minus(119909 sin 120579119894 cos120593119894+119910 sin 120579119894 sin120593119894+ cos 120579119894)The wave vector is

997888

119896 = 120573119896 and phase shift

997888

11989611990500 =

997888

119896 |119905 The

array is assumed to be periodic in the 119909119910-plane and the cell ofthe array is obtained by shifting the cell through the relation997888120588119898119899= 119898sdot

997888119886 1+119899sdot

997888119886 2

997888120588119898119899

is defined as the translation vectorof the lattice

21 Integral Equation In this paper we consider the metallicphotonic crystal Let 119878 be the surface of a metallic object Byenforcing the boundary conditions that the total tangentialelectric field is zero on the perfect electric conductor (PEC)surface 119878 the electric field integral equation (EFIE) is givenas [13]

119895120596120583 [int[

997888

119869119904(997888119903

1015840

) +

11205732nabla1015840sdot

997888

119869119904(997888119903

1015840

)nabla]

sdot 119866119901(997888119903 997888119903

1015840

) 1198891199041015840]

119905

=

997888

119864

inc119905

(1)

where 119866119901is a doubly periodic Greenrsquos function The Rao-

Wilton-Glisson (RWG) basis functions on triangular ele-ments are employed to discretize the electric current [14] Inaddition the special basis and test functions derived fromRWG basis function are introduced to ensure current con-tinuity across the periodic boundaries [15] The applicationof the Floquet-Bloch theorem reduces the computational

x

y

zPlane wave120579

120593

a1

a2

Figure 1 Multilayered periodic structure with a general skewedlattice

domain of infinite periodic structures to a single unit cell butleads to the numerical evaluation of very slowly convergingseries

22 Modified Ewald Transformation We propose the follow-ing strategy to compute the spatial periodic Greenrsquos functionin an arbitrary point 997888119906 = 997888119903 minus 997888119903

1015840

If 119906119911= |119911 minus 119911

1015840| gt 05radic119860 PGF is computed by its spectral

representation with exponential convergence and thus noacceleration technique is required [16] Consider

119866119901(997888119903 997888119903

1015840

) =

infin

sum

119898=minusinfin

infin

sum

119899=minusinfin

119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840

)

2119895119860119896119911119898119899

119890minus119895119896119911119898119899|119911minus1199111015840| (2)

where

997888119903 =

997888120588 +119911 sdot

997888

119896119905119898119899

=

997888

11989611990500 +

2120587119860

[119898 (997888119886 2 times ) + 119899 ( times

997888119886 1)]

(3)

is the reciprocal lattice vector Consider

119896119911119898119899

=radic11989620 minus

997888

119896119905119898119899

sdot

997888

119896119905119898119899

(4)

where Re(119896119911119898119899

) ge 0 Im(119896119911119898119899

) le 0If 119906119911= |119911 minus 119911

1015840| lt 05radic119860 PGF is computed by the

representation of Ewald transformation [17] which are twoexponential converging series Consider

119866119901(997888119903 997888119903

1015840

) = 1198661199011 (

997888119903 997888119903

1015840

) +1198661199012 (

997888119903 997888119903

1015840

) (5)


Primary source

Field

Image sources

Image sources

(a)

(minus1 0) (0 0) (1 0)


(minus1 1) (0 1) (1 1)

(b)


where

1198661199011 (

997888119903 997888119903

1015840

) =

infin

sum

119898=minusinfin

infin

sum

119899=minusinfin

119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840

)

4119895119860119896119911119898119899


119895119896119911119898119899

2119864minus (119911 minus 119911

1015840) 119864)

+ 119890119895119896119911119898119899(119911minus1199111015840) erfc(

119895119896119911119898119899

2119864+ (119911 minus 119911

1015840) 119864)]

(6)

1198661199012 (

997888119903 997888119903

1015840

) =

infin

sum

119898=minusinfin

infin

sum

119899=minusinfin

119890minus11989599788811989611990500 sdot997888120588119898119899

4120587119877119898119899

sdot real [1198901198951198960119877119898119899 erfc(119877119898119899119864+

119895119896

2119864)]

(7)


119898119899= |997888119903 minus

997888119903

1015840

minus997888120588119898119899|


119864 = max(radic 120587

119860

119896

2119867) (8)



1015840| gt







119911is too



1015840





1015840

|119899(119899 ge minus3) type




119866001199012 (

997888119903 997888119903

1015840

) = 119865001199012 (

997888119903 997888119903

1015840

) +

14120587119877

(9)


o

a2

a1

Gn119909n119910n119911Gn119909+1n119910n119911

Gn119909+1n119910+1n119911Gn119909n119910+1n119911

Gn119909+1n119910n119911+1

Gn119909n119910+1n119911+1

Gn119909n119910n119911+1

Gn119909+1n119910+1n119911+1

dx

dydz

G(u)


where 119865001199012


119865001199012 (

997888119903 997888119903

1015840

) =

14120587119877


)minus erfc(minus11989511989602119864

)

(10)


lim119877rarr 0

119865001199012 (

997888119903 997888119903

1015840

)

=

12120587


)minus

2119864radic120587

119890119896204119864

2]

(11)


119866(997888119906) = (1minus120572

119909) (1minus120572

119910) (1minus120572

119911) 119866119899119909+1119899119910+1119899119911+1

+ (1minus120572119909) (1minus120572

119910) 120572119911119866119899119909+1119899119910+1119899119911+2

+ (1minus120572119909) 120572119910(1minus120572

119911) 119866119899119909+1119899119910+2119899119911+1

+ (1minus120572119909) 120572119910120572119911119866119899119909+1119899119910+2119899119911+2

+120572119909(1minus120572

119910) (1minus120572

119911) 119866119899119909+2119899119910+1119899119911+1

+120572119909(1minus120572

119910) 120572119911119866119899119909+2119899119910+1119899119911+2

+120572119909120572119910(1minus120572

119911) 119866119899119909+2119899119910+2119899119911+1

+120572119909120572119910120572119911119866119899119909+2119899119910+2119899119911+2

(12)

where 120572119909= 119889119909119871 120572119910= 119889119910119871 120572119911= 119889119911119871


120576 =

100381610038161003816100381610038161003816100381610038161003816

119866intminus 119866

ex

119866ex

100381610038161003816100381610038161003816100381610038161003816

(13)





3 ACA Technique



119898times119899by




o

B

C

D

AB

CD

A

10 20 30 40 50 60 70 80 90 100 1101E minus 7

1E minus 6

1E minus 5

1E minus 4

1E minus 3

Rela

tive e

rror

120576Nint

a1

a2


9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896

119909=





119885119898times119899

= 119880119898times119903

119881119903times119899

=

119903

sum

119894=1119906119894

119898times1V119894

119899times1 (14)



119880119898times119903

and 119881119903times119899




minus119885119898times119899

10038171003817100381710038171003817119865le 120576

1003817100381710038171003817119885119898times119899

1003817100381710038171003817119865

(15)











120578 () = (1minus 119903 sdot (119898 + 119899)

119898 sdot 119899

) sdot 100 (16)





4 Numerical Results



1= 119909 6mm and 997888119886

2= 119910 6mm Each





minus50

minus40

minus30

minus20

minus10

0

Refl

and

Tran

s co

effici

ents

(dB)

Frequency (GHz)


o

1

2

Unit cell

10 50403020

a1

a2


20 4540353025minus80

minus60

minus40

minus20

0

Tran

simiss

ion

(dB)

Frequency (GHz)


Layer

1

2

3

N minus 1

N





10 201816141200

02

04

06

08

10

Refle

ctio

n

4-layer 8-layer


w

d

h

120582 (120583m)



11988511

11988512

11988513

11988514

11988515

11988516

11988517

mdash 2078 864 540 594 567 48611988518

11988519

119885110

119885111

119885112

119885113

119885114

621 567 405 351 378 459 432119885115

119885116

119885117

119885118

119885119

119885120

119885121




5 Conclusion






Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Primary source

Field

Image sources

Image sources

(a)

(minus1 0) (0 0) (1 0)


(minus1 1) (0 1) (1 1)

(b)


where

1198661199011 (

997888119903 997888119903

1015840

) =

infin

sum

119898=minusinfin

infin

sum

119899=minusinfin

119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840

)

4119895119860119896119911119898119899


119895119896119911119898119899

2119864minus (119911 minus 119911

1015840) 119864)

+ 119890119895119896119911119898119899(119911minus1199111015840) erfc(

119895119896119911119898119899

2119864+ (119911 minus 119911

1015840) 119864)]

(6)

1198661199012 (

997888119903 997888119903

1015840

) =

infin

sum

119898=minusinfin

infin

sum

119899=minusinfin

119890minus11989599788811989611990500 sdot997888120588119898119899

4120587119877119898119899

sdot real [1198901198951198960119877119898119899 erfc(119877119898119899119864+

119895119896

2119864)]

(7)


119898119899= |997888119903 minus

997888119903

1015840

minus997888120588119898119899|


119864 = max(radic 120587

119860

119896

2119867) (8)



1015840| gt







119911is too



1015840





1015840

|119899(119899 ge minus3) type




119866001199012 (

997888119903 997888119903

1015840

) = 119865001199012 (

997888119903 997888119903

1015840

) +

14120587119877

(9)


o

a2

a1

Gn119909n119910n119911Gn119909+1n119910n119911

Gn119909+1n119910+1n119911Gn119909n119910+1n119911

Gn119909+1n119910n119911+1

Gn119909n119910+1n119911+1

Gn119909n119910n119911+1

Gn119909+1n119910+1n119911+1

dx

dydz

G(u)


where 119865001199012


119865001199012 (

997888119903 997888119903

1015840

) =

14120587119877


)minus erfc(minus11989511989602119864

)

(10)


lim119877rarr 0

119865001199012 (

997888119903 997888119903

1015840

)

=

12120587


)minus

2119864radic120587

119890119896204119864

2]

(11)


119866(997888119906) = (1minus120572

119909) (1minus120572

119910) (1minus120572

119911) 119866119899119909+1119899119910+1119899119911+1

+ (1minus120572119909) (1minus120572

119910) 120572119911119866119899119909+1119899119910+1119899119911+2

+ (1minus120572119909) 120572119910(1minus120572

119911) 119866119899119909+1119899119910+2119899119911+1

+ (1minus120572119909) 120572119910120572119911119866119899119909+1119899119910+2119899119911+2

+120572119909(1minus120572

119910) (1minus120572

119911) 119866119899119909+2119899119910+1119899119911+1

+120572119909(1minus120572

119910) 120572119911119866119899119909+2119899119910+1119899119911+2

+120572119909120572119910(1minus120572

119911) 119866119899119909+2119899119910+2119899119911+1

+120572119909120572119910120572119911119866119899119909+2119899119910+2119899119911+2

(12)

where 120572119909= 119889119909119871 120572119910= 119889119910119871 120572119911= 119889119911119871


120576 =

100381610038161003816100381610038161003816100381610038161003816

119866intminus 119866

ex

119866ex

100381610038161003816100381610038161003816100381610038161003816

(13)





3 ACA Technique



119898times119899by




o

B

C

D

AB

CD

A

10 20 30 40 50 60 70 80 90 100 1101E minus 7

1E minus 6

1E minus 5

1E minus 4

1E minus 3

Rela

tive e

rror

120576Nint

a1

a2


9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896

119909=





119885119898times119899

= 119880119898times119903

119881119903times119899

=

119903

sum

119894=1119906119894

119898times1V119894

119899times1 (14)



119880119898times119903

and 119881119903times119899




minus119885119898times119899

10038171003817100381710038171003817119865le 120576

1003817100381710038171003817119885119898times119899

1003817100381710038171003817119865

(15)











120578 () = (1minus 119903 sdot (119898 + 119899)

119898 sdot 119899

) sdot 100 (16)





4 Numerical Results



1= 119909 6mm and 997888119886

2= 119910 6mm Each





minus50

minus40

minus30

minus20

minus10

0

Refl

and

Tran

s co

effici

ents

(dB)

Frequency (GHz)


o

1

2

Unit cell

10 50403020

a1

a2


20 4540353025minus80

minus60

minus40

minus20

0

Tran

simiss

ion

(dB)

Frequency (GHz)


Layer

1

2

3

N minus 1

N





10 201816141200

02

04

06

08

10

Refle

ctio

n

4-layer 8-layer


w

d

h

120582 (120583m)



11988511

11988512

11988513

11988514

11988515

11988516

11988517

mdash 2078 864 540 594 567 48611988518

11988519

119885110

119885111

119885112

119885113

119885114

621 567 405 351 378 459 432119885115

119885116

119885117

119885118

119885119

119885120

119885121




5 Conclusion






Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










o

a2

a1

Gn119909n119910n119911Gn119909+1n119910n119911

Gn119909+1n119910+1n119911Gn119909n119910+1n119911

Gn119909+1n119910n119911+1

Gn119909n119910+1n119911+1

Gn119909n119910n119911+1

Gn119909+1n119910+1n119911+1

dx

dydz

G(u)


where 119865001199012


119865001199012 (

997888119903 997888119903

1015840

) =

14120587119877


)minus erfc(minus11989511989602119864

)

(10)


lim119877rarr 0

119865001199012 (

997888119903 997888119903

1015840

)

=

12120587


)minus

2119864radic120587

119890119896204119864

2]

(11)


119866(997888119906) = (1minus120572

119909) (1minus120572

119910) (1minus120572

119911) 119866119899119909+1119899119910+1119899119911+1

+ (1minus120572119909) (1minus120572

119910) 120572119911119866119899119909+1119899119910+1119899119911+2

+ (1minus120572119909) 120572119910(1minus120572

119911) 119866119899119909+1119899119910+2119899119911+1

+ (1minus120572119909) 120572119910120572119911119866119899119909+1119899119910+2119899119911+2

+120572119909(1minus120572

119910) (1minus120572

119911) 119866119899119909+2119899119910+1119899119911+1

+120572119909(1minus120572

119910) 120572119911119866119899119909+2119899119910+1119899119911+2

+120572119909120572119910(1minus120572

119911) 119866119899119909+2119899119910+2119899119911+1

+120572119909120572119910120572119911119866119899119909+2119899119910+2119899119911+2

(12)

where 120572119909= 119889119909119871 120572119910= 119889119910119871 120572119911= 119889119911119871


120576 =

100381610038161003816100381610038161003816100381610038161003816

119866intminus 119866

ex

119866ex

100381610038161003816100381610038161003816100381610038161003816

(13)





3 ACA Technique



119898times119899by




o

B

C

D

AB

CD

A

10 20 30 40 50 60 70 80 90 100 1101E minus 7

1E minus 6

1E minus 5

1E minus 4

1E minus 3

Rela

tive e

rror

120576Nint

a1

a2


9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896

119909=





119885119898times119899

= 119880119898times119903

119881119903times119899

=

119903

sum

119894=1119906119894

119898times1V119894

119899times1 (14)



119880119898times119903

and 119881119903times119899




minus119885119898times119899

10038171003817100381710038171003817119865le 120576

1003817100381710038171003817119885119898times119899

1003817100381710038171003817119865

(15)











120578 () = (1minus 119903 sdot (119898 + 119899)

119898 sdot 119899

) sdot 100 (16)





4 Numerical Results



1= 119909 6mm and 997888119886

2= 119910 6mm Each





minus50

minus40

minus30

minus20

minus10

0

Refl

and

Tran

s co

effici

ents

(dB)

Frequency (GHz)


o

1

2

Unit cell

10 50403020

a1

a2


20 4540353025minus80

minus60

minus40

minus20

0

Tran

simiss

ion

(dB)

Frequency (GHz)


Layer

1

2

3

N minus 1

N





10 201816141200

02

04

06

08

10

Refle

ctio

n

4-layer 8-layer


w

d

h

120582 (120583m)



11988511

11988512

11988513

11988514

11988515

11988516

11988517

mdash 2078 864 540 594 567 48611988518

11988519

119885110

119885111

119885112

119885113

119885114

621 567 405 351 378 459 432119885115

119885116

119885117

119885118

119885119

119885120

119885121




5 Conclusion






Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










o

B

C

D

AB

CD

A

10 20 30 40 50 60 70 80 90 100 1101E minus 7

1E minus 6

1E minus 5

1E minus 4

1E minus 3

Rela

tive e

rror

120576Nint

a1

a2


9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896

119909=





119885119898times119899

= 119880119898times119903

119881119903times119899

=

119903

sum

119894=1119906119894

119898times1V119894

119899times1 (14)



119880119898times119903

and 119881119903times119899




minus119885119898times119899

10038171003817100381710038171003817119865le 120576

1003817100381710038171003817119885119898times119899

1003817100381710038171003817119865

(15)











120578 () = (1minus 119903 sdot (119898 + 119899)

119898 sdot 119899

) sdot 100 (16)





4 Numerical Results



1= 119909 6mm and 997888119886

2= 119910 6mm Each





minus50

minus40

minus30

minus20

minus10

0

Refl

and

Tran

s co

effici

ents

(dB)

Frequency (GHz)


o

1

2

Unit cell

10 50403020

a1

a2


20 4540353025minus80

minus60

minus40

minus20

0

Tran

simiss

ion

(dB)

Frequency (GHz)


Layer

1

2

3

N minus 1

N





10 201816141200

02

04

06

08

10

Refle

ctio

n

4-layer 8-layer


w

d

h

120582 (120583m)



11988511

11988512

11988513

11988514

11988515

11988516

11988517

mdash 2078 864 540 594 567 48611988518

11988519

119885110

119885111

119885112

119885113

119885114

621 567 405 351 378 459 432119885115

119885116

119885117

119885118

119885119

119885120

119885121




5 Conclusion






Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus50

minus40

minus30

minus20

minus10

0

Refl

and

Tran

s co

effici

ents

(dB)

Frequency (GHz)


o

1

2

Unit cell

10 50403020

a1

a2


20 4540353025minus80

minus60

minus40

minus20

0

Tran

simiss

ion

(dB)

Frequency (GHz)


Layer

1

2

3

N minus 1

N





10 201816141200

02

04

06

08

10

Refle

ctio

n

4-layer 8-layer


w

d

h

120582 (120583m)



11988511

11988512

11988513

11988514

11988515

11988516

11988517

mdash 2078 864 540 594 567 48611988518

11988519

119885110

119885111

119885112

119885113

119885114

621 567 405 351 378 459 432119885115

119885116

119885117

119885118

119885119

119885120

119885121




5 Conclusion






Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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